Shown in FIG. 1 is a side elevation view of a crank, pedal and chain-sprocket drive in a conventional arrangement as typically used for manually powering a bicycle. FIG. 2 is a partial section elevation view of the assembly of FIG. 1 taken generally along the section line II--II of FIG. 1, and in the direction of the arrows.
As illustrated, chain-sprocket 11 has a working diameter D1, and is typically configured for No. 41 roller chain, although other sizes have been and are used for bicycle propulsion. Crank member 13 has a horizontal portion 29 that extends through a frame 25, and is engaged in the frame by means of ball bearings 27 and 35. Details of machining of races and shoulders on both crank portion 29 and the inside of the bore through frame 25 to effect the mounting of the ball bearings are not shown.
The crank has a flanged portion 21 near one end of horizontal portion 29, and sprocket 11 is fixedly attached to flange 21 by means of fasteners such as fastener 15. The plane of flange 21, and hence the plane of chain-sprocket 11, are at right angles to portion 29 of crank 13.
Beyond the bearing mounting and the sprocket mounting, crank 13 turns at right angles to portion 29 and forms a crank arm of length L ending in a pivot mounting 19 to which a pedal assembly 17 is rotationally attached. At the other side of frame 25 the crank makes a similar bend, also at right angles to portion 29, but in the direction opposite the first crank arm. The second crank arm is also of length L and ends in a secondary rotationally mounted pedal assembly 31. The crank arm on the sprocket side has a projection 33 which extends through an opening in sprocket 11 and helps to transfer torque from the crank to the sprocket in use, so that all the torque is not transmitted through the screw fasteners, such as fastener 15, that attach the sprocket to the crank. In some crank and sprocket arrangements the projection to the sprocket fastens to the sprocket by conventional fasteners, and in some no such projection is used. Also, the sprocket and crank may be a single unit formed from a machined casting, or by other means. In many arrangements there is a single sprocket, and in many arrangements there are multiple sprockets of different diameters mounted side by side with a deraileur mechanism operable by the rider of the bicycle to move the drive chain from one sprocket to another to change the mechanical advantage of the drive train.
The arrangement of the crank, pedals and sprocket is such, as is best illustrated by FIG. 1, that the rider of the bicycle may exert force in a generally downward direction on the pedal at the end of one extension of the crank, and that force, represented by force F1 in FIG. 1, will be translated to a varying torque on chain-sprocket 11 as the crank rotates and the bicycle is propelled forward. The torque exerted on sprocket 11 by the crank is converted to a force in an endless chain (not shown) which becomes a torque in a rear sprocket (also not shown) which drives a rear wheel of the bicycle.
Rotation of crank 13 is shown in FIG. 1 beginning arbitrarily as zero rotation with the crank arm on the sprocket side extending vertically upward. Rotation proceeds, also arbitrarily, in the direction of arrow 23, and the amount of rotation is shown in radians for each one-quarter of a revolution through a first full revolution. The crank arm on the sprocket side is shown in FIG. 1 at an angle O from the vertical, and O is meant to represent any rotation from zero to one-half revolution. When the sprocket-side crank arm has revolved through one-half revolution, and points vertically downward, the opposite side crank arm (ending in pedal 31) will point vertically upward, and the bicycle rider then typically shifts his force to this opposite pedal for the next one-half revolution.
It is quite true that a rider may exert more or less force on a pedal, and the force may not necessarily be directed exactly vertically downward. It is also true that the force exerted over a half-revolution will vary both in direction and amplitude, and there may be many force patterns exerted by a rider. For purposes of calculation and comparison, however, it is convenient to consider the force as a constant force exerted vertically downward.
To understand the operation of the conventional pedal and crank arrangement of the prior art, it is convenient to apply engineering and mathematical concepts of force, work and power. Force F1 is exerted on pedal 17 in FIG. 1, and force is transferred to the crank at rotational connection 19. The force on the crank arm may be resolved by the methods of vectorial combination into two forces, one in a direction along the axis of the crank arm and the other at right angles to the crank arm. It may be shown that at .theta.=0, with the crank arm vertically upward, force F1, which is shown vertically downward, will have no component at right angles to the crank, and the entire force will be along the crank axis toward the frame. This condition is shown by FIG. 3A. As the crank rotates, .theta. becomes greater than zero, and FIG. 3B shows a force vector diagram for an approximate angular rotation of 30 degrees. At this position of angular rotation, force F1 has a component F2 at right angles to crank 13, and a component F3 along the axis of the crank arm toward the frame. Force F3, passing through the center of rotation of the crank, provides no rotational torque to the crank and sprocket assembly. It may be shown that the instant force F3, at the position shown, has no propulsive effect on the bicycle, even though force F3 may have a component in the direction of movement of the bicycle (or in the opposite direction). This is so, because for the rider to exert this force on the frame, an equal and opposite force must be exerted elsewhere on the frame, or the rider would else not move along with the bicycle. The instant force F2, at right angles to the crank arm, is actually exerted at the rotational attachment point 19, and provides a rotational couple to the crank equal in magnitude to F2 * L.
As rotation continues, F1 remaining consant in magnitude and downward direction, F2 perpendicular to the crank arm increases in magnitude and F3 along the crank arm axis toward the frame decreases in magnitude, until, at 90 degrees rotation (.theta.=.pi./2), F3 becomes zero, and F2 becomes F1, as is shown in FIG. 3C. At this point in rotation the torque applied to the crank and sprocket assembly is maximum at F1 * L.
When rotation of the crank is beyond the 90 degree point, component F2 perpendicular to the crank arm begins to decrease from the maximum magnitude, and F3 reappears, but in the direction along the crank arm axis away from the frame. The situation at .theta.=3.pi./4 (135 degrees) is shown in FIG. 3D. Force F2, which is in reality applied at rotational point 19, provides an instantaneous torque equal to F2 * L, and the force F3, which passes through the axis of rotation of the crank and sprocket assembly, provides, as before, no propulsive effect. At .theta.=.pi.(180 degrees), which is one-half revolution of the crank and sprocket assembly, shown in FIG. 3E, the entire force F1 is along the axis of the crank, away from the frame, and there is no component at right angles to the crank arm. There is therefore no propulsive torque applied at this point.
At the point shown in FIG. 3E, one-half revolution, the opposite side crank arm is vertically upward, and the rider shifts his application of force to that side. The force components and torque effects for the second half revolution follow exactly the diagrams presented as FIGS. 3A and 3E.
Mathematically, the general case can be expressed from FIG. 3B, with .theta. representing any angle of rotation from 0 degrees through (and including) 180 degrees (one-half revolution, for which .theta.=.pi.). Of primary interest is the magnitude of F2, which determines the magnitude of torque applied to the crank arrangement. The vector right triangle formed by the force vectors has F1 as hypotenuse, and angle .theta. is the angle between F1 and F3. By definition of the Sine function in trigonometry, F2=F1*Sin.theta.. As .theta. increases from 0 degrees, Sin.theta. increases from zero to a maximum value 1 at .theta.=90 degrees (.pi./2 radians). So at 90 degrees F2=F1, and F3 disappears.
The case for the second quarter revolution from .theta.=90 degrees to .theta.=180 degrees can be expressed from FIG. 3D, for which .theta. is between these two values. In FIG. 3D .theta. is greater than 90 degrees, and is therefore larger than any of the internal angles of the vector right triangle of F1, F2 and F3. Considering the vector right triangle, again by the definition of the Sine function in trigonometry, the force F2 perpendicular to the crank arm can be expressed as a function of the applied force F1, as F2=F1*Sin.beta.. The angle .beta. is the angle between the vectors F1 and F3. By trigonometry it can be shown that .beta.=180 degrees-.theta., and that Sin(180 degrees-.theta.) is always equal to Sin.theta.. Therefore, for rotation between 90 degrees and 180 degrees, F2=F1*Sin.theta., just as was true for rotation between zero and 90 degrees. The relationship F2=F1*Sin.theta. therefore holds for all points of rotation between zero and 180 degrees, and the torque applied to the sprocket for all such points of rotation of one arm of the crank is T=F1*L*Sin.theta..
Since the force and torque analysis for the opposite crank arm to which force F1 is shifted as .theta. reaches 180 degrees is identical to the analysis just done for the first crank arm, the torque applied for the second half revolution of the crank follows the same relationship as for the first half revolution. The Sine function is a harmonic function related to revolution. A plot of the torque producing force F2, or of torque (since L is constant) applied to the sprocket by application of a constant force F1 downward on the pedal of one crank arm during rotation of that crank arm between zero and 180 degrees, then transferring that force to the opposite crank arm for its rotation between zero and 180 degrees, the two half revolutions making one complete revolution of the sprocket, is approximately as shown in FIG. 4. Torque or Force F2 is the ordinate (vertical axis), and revolutions of the sprocket is the abcissa (horizontal axis). For each half revolution of the sprocket the torque or force describes the positive half of a full revolution Sine function. The function reaches zero for each half revolution, but never goes negative (as does a true Sine function) because the driving force F1 is shifted each half revolution to the opposite pedal, keeping the torque positive.
The graph of FIG. 4 showing the force and torque variations for the idealized case as a function of sprocket revolution can also be used to illustrate the work done by a rider in propelling the bicycle. The engineering definition of work is: The product of a force exerted on a body and displacement of the body in the direction of the force. In the present case, the force which does work is the force F2 directed at right angles to one or the other of the two arms of the crank. Rotation of the crank and sprocket assembly around the rotational axis in the frame bearings provides movement of the point of application of the force F2 always in a circular arc at a distance L from the center of rotation, which is the length of either crank arm, and is always in the direction of application of Force F2.
Applying the concepts of differential and integral calculus, in any instant of rotation through an infinitesimal angle d.theta., the instant variable force, F2, is applied to the crank arm at point 19; and the distance along the curved path that point 19 moves relative to the rotational center is Ld.theta., because the length of an arc is the radius of the arc times the angular revolution in radians. The differential (instantaneous) work done is therefore: EQU dW=F2*Ld.theta.
Since the variable force F2 can be expressed as a function of the applied force F1, as F2=F1*Sin.theta., the differential expression for the work done becomes: EQU dW=F1*L*Sin.theta.*d.theta.
Integration applied to determine the work done between .theta.=0 and .theta.=1/2 revolution then yields: ##EQU1## or W=F1*L*{[-Cos.theta.].theta.=.pi./2-[-Cos.theta.].theta.=0}for which W=F1*L*{[-(-1)]-[0]}=F1*L*(1-0) i.e. W=F1*L for one half revolution of the sprocket. It follows therefore, that for the idealized case, the work for a full revolution of the sprocket is 2F1*L; and, if N is the number of sprocket revolutions, the total work done through multiple revolutions will be: EQU W.sub.T =2N*F1*L
In the graph of FIG. 4, since the torque multiplied by the angular travel is the same as the force F2 multiplied by the arc length, any very small area under the curve, such as blackened area 37, may represent the differential F1*L*Sin.theta.*d.theta., with the height D2 equal to F1*L*Sin.theta., and the width D3 equal to the vanishingly small d.theta.. The value of the definite integral between .theta.=0 and .theta.=.pi./2, which is the work done for 1/2 revolution, is the entire shaded area under the torque curve bounded by the curve and the zero torque axis, denoted area 39 in FIG. 4.
As an example, if a rider provides a downward constant force of 20 pounds (9.072 Kg.) on the pedals, with a crank length of 9 inches (0.75 ft., 22.86 cm.), and the rider travels a distance requiring 1000 revolutions of the sprocket, the work done will be: EQU W=2*1000*20*.75=30,000 Ft-Lbs. EQU or PST/W=4150 KG-M=40,680 Joules
Power is the rate of doing work. If the rate in the example is one revolution of the crank per second, the 1000 revolutions are accomplished in 1000 seconds with 30,000 Ft-Lbs of work performed. The power is 30 Ft-Lbs/sec., which is 0.055 Horsepower.
For the purpose of this specification a power stroke is defined as the application of a downward force by a rider on a pedal at the end of one crank arm over the time for the crank arm to revolve from a vertical up position to a vertical down position. The variations in moment force F2 and torque over that period, assuming the force applied is vertically downward and constant over the period, are shown in the part of the graph of FIG. 4 from 0 revolutions of the crank to 1/2 revolution. The work done over that period is represented by the shaded area under the curve, and has been shown to be equal to F1*L. It may also be shown mathematically that if all the applied force (F1) over a power stroke were convertible to useful work, the work would be F1*.pi.*L, and .pi.*L in this expression is equal to the arc length described by the outer end of a crank arm at distance L from the rotational center in the frame, over the 1/2 revolution of the power stroke. The ratio of the theoretical maximum work to the actual is the mathematical constant .pi., which is approximately 3.14159. The actual work is 1/.pi., or about 31.8% of the theoretical maximum work. The concept of the power stroke is convenient, because it repeats for each 1/2 revolution of the crank and sprocket, and the total work over any period can be related to the number of power strokes.
One clear difficulty with the conventional crank and sprocket arrangement is the fact that not all of the force applied by a rider is convertible to useful work. As is evident by the force analysis presented above, including the graph of FIG. 4, the force doing work is only equal to the maximum force applied to the pedals by a rider at one instant in the power stroke, when the crank arm is horizontal as shown in FIG. 3C, and F2=F1. At all other times F2 is less than F1. As a result, much effort may be wasted by a rider, particularly near the beginning and end of a power stroke, when most of the force applied by the rider is absorbed by the frame, and only a small portion goes into producing torque to propel the bicycle. Clearly what is needed is a mechanism to make more use of the forces a rider may apply to propel the bicycle.